Fractal structure and its forming method

ABSTRACT

A fractal structure is formed to have a plurality of regions different in fractal dimension characterizing the self-similarity. Especially in a stellar fractal structure, a region with a low fractal dimension is formed around a core with a high fractal dimension. By adjusting the ratios in volume of these regions relative to the entire fractal structure, the nature of phase transition occurring in the fractal structure, such as a magnetization curve of Mott transition or ferromagnetic phase transition, quantum chaos in the electron state, or the like. For enhancing the controllability, the fractal dimension of the core is preferably larger than 2.7 and the fractal dimension of the region around the core is preferably smaller than 2.3.

TECHNICAL FIELD

[0001] This invention relates a fractal structure and its formingmethod, especially based on a novel principle.

BACKGROUND ART

[0002] For application of a solid material to electronic or opticaldevices, physical properties of the material may restrict itsapplications. For example, in case of using a semiconductor material ina light emitting device, it will be usable in a device of an emissionwavelength corresponding to the band gap of the material, but someconsideration will be necessary for changing the emission wavelength.Regarding physical properties related to semiconductor bands, controlsby superlattices have been realized. More specifically, by changing theperiod of a superlattice, the bandwidth of its subband can be controlledto design an emission wavelength.

[0003] Targeting on controlling many-electron-state structures bymaterial designs, the Inventor proposed many-body effect engineering byquantum dot-based structures and has continued theoretical analyses ((1)U.S. Pat. No. 5,430,309; (2) U.S. Pat. No. 5,663,571; (3) U.S. Pat. No.5,719,407; (4) U.S. Pat. No. 5,828,090; (5) U.S. Pat. No. 5,831,294; (6)U.S. Pat. No. 6,020,605; (7) J. Appl. Phys. 76, 2833(1994); (8) Phys.Rev. B51, 10714(1995); (9) Phys. Rev. B51, 11136(1995); (10) J. Appl.Phys. 77, 5509(1995); (11) Phys. Rev. B53, 6963(1996); (12) Phys. Rev.B53, 10141(1996); (13) Appl. Phys. Lett. 68, 2657(1996); (14) J. Appl.Phys. 80, 3893(1996); (15) J. Phys. Soc. Jpn. 65, 3952(1996); (16) Jpn.J. Appl. Phys. 36, 638(1997); (17) J. Phys. Soc. Jpn. 66, 425(1997);(18) J. Appl. Phys. 81, 2693 (1997); (19) Physica (Amsterdam) 229B,146(1997); (20) (Physica (Amsterdam) 237A, 220(1997); (21) Surf. Sci.375, 403(1997); (22) Physica (Amsterdam) 240B, 116(1997); (23) Physica(Amsterdam) 240B, 128(1997); (24) Physica (Amsterdam) IE, 226(1997);(25) Phys. Rev. Lett. 80, 572(1998); (26) Jpn. J. Appl. Phys. 37,863(1998); (27) Physica (Amsterdam) 245B, 311(1998); (28) Physica(Amsterdam) 235B, 96(1998); (29) Phys. Rev. B59, 4952(1999); (30) Surf.Sci. 432, 1(1999); (31) International Journal of Modern Physics B. Vol.13, No. 21, 22, pp. 2689-2703, 1999). For example, realization ofvarious correlated electronic systems is expected by adjusting atunneling phenomenon between quantum dots and interaction betweenelectrons in quantum dots. Let the tunneling transfer between adjacentquantum dots be written as t. Then, if quantum dots are aligned in formof a tetragonal lattice, the bandwidth of one electron state isT_(eff)=4t. If quantum dots form a one-dimensional chain, the bandwidthof one electron state is T_(eff)=2t. In case of a three-dimensionalquantum dot array, T_(eff)=6t. That is, if D is the dimension of aquantum dot array, the bandwidth of one electron state has beenT_(eff)=2Dt. Here is made a review about half-filled (one electron pereach quantum dot) Mott transition (also called Mott-Hubbard transitionor Mott metal-insulator transition). Let the effective interaction ofelectrons within a quantum dot be written as U_(eff), then the Hubbardgap on the part of the Mott insulator is substantially described asΔ=U_(eff)−T_(eff), and the Mott transition can be controlled by changingU_(eff) or t. As already proposed, the Mott-Hubbard transition can becontrolled by adjusting U_(eff) or t, using a field effect, and it isapplicable to field effect devices (Literatures (5), (6), (11) and (14)introduced above).

[0004] On the other hand, reviewing the equation ofΔ=U_(eff)−T_(eff)=U_(eff)−2Dt, it will be possible to controlMott-Hubbard transition by controlling the dimensionality D of thesystem. For this purpose, the Applicant already proposed a fractal-basedstructure that can continuously change the dimensionality, and haveexhibited that Mott-Hubbard transition is controllable by changing thefractal dimensions.

[0005] To enable designing of wider materials, it is desired to modifyand control the dimension of materials by designing methods beyond thesimple fractal nature.

[0006] As physical systems in charge of information processing,intrinsic non-linearity is indispensable. As devices having been usedfor years, there are electronic devices using materials exhibitingnon-linear responses to certain extents. For example, two-terminaldevices exhibiting differential negative resistances are one example ofdevices that are non-linear in current-voltage characteristics. Ofcourse, MOS-FET, which is a three-terminal device, supports the moderntechnologies. By coupling these electronic devices having non-linearproperties with a linear electronic circuit and constructing aninformation processing device having a non-linear property, any desiredcalculation can be executed.

[0007] However, difficulties by high integration have become issues withsuch electronic circuits. Generation of heat is one of the problems, forexample. The heat generation caused by intrinsic electric resistance ismandatory for producing non-linearity of an electronic device, andindispensable and essential for executing information processing.

[0008] To avoid this difficulty, trials have been made to decrease thenumber of devices by enhancing non-linearity of component devices.Progress of this scheme necessarily results in the need for componentdevices having non-linearity as strong as chaotic. When a chaoticclassical system is quantized, what characterizes the behaviors of thequantum system is quantum chaos.

[0009] On the other hand, as miniaturization of component devicesprogresses, electrons confined in a device result in behaving as quantummechanical particles. Therefore, from this point of view, hopes areplaced on components devices exhibiting quantum chaos. The Applicant hascontinued to theoretically demonstrate that, in a quantum system in astructure having a fractal configuration, the quantum chaos can becontrolled by changing the fractal dimension characterizing the system.

[0010] An object the invention intends to accomplish is to provide afractal structure and its forming method capable of modulating andcontrolling the dimensionality of a material by a design method beyondthe simple fractal nature.

[0011] Another object the invention intends to accomplish is to providea fractal structure and its forming method capable of controlling phasetransition and chaos, in particular, quantum chaos, by a design methodbeyond the simple fractal nature.

DISCLOSURE OF INVENTION

[0012] The Inventor found, through concentrated researches towardsolution of those issues, that a more complex fractal structure having aportion characterized by a plurality of fractal dimensions can be formedby changing the growth conditions with time during growth of the fractalstructure. Especially in the process of growing a random fractal, it hasbeen found that a fractal structure in form of nerve cells can be formedby forming a region with a low fractal dimension around a core with ahigh fractal dimension. Then, it has been found that, in fractalstructures of this type, occurrence of phase transition, such asmagnetic phase transition, and chaos, such as quantum chaos in anelectron state, can be controlled. As a result of later detailedanalysis, it has been found that there are fractal dimensions suitablefor controlling these phenomena.

[0013] The present invention has been made on the basis of thoseresearches by the Inventor.

[0014] That is, to overcome the above-indicated problems, according tothe first aspect of the invention, there is provided a fractal structurecomprising a plurality of regions different in fractal dimensioncharacterizing the self-similarity.

[0015] In the first aspect of the invention, nature of phase transitionoccurring in a fractal structure is controlled by adjusting the ratiosin volume of a plurality of regions relative to the entire fractalstructure. Alternatively, correlation between interactive electrons inan electron system is controlled. Additionally, the magnetization curveof ferromagnetic phase transition is controlled. Further, nature of thechaos appearing in the fractal structure is controlled, morespecifically, the quantum chaos in electrons states, for example, iscontrolled. Control of the quantum chaos in electron states can becontrolled with a high controllability by using introduction of a randommagnetic field by addition of a magnetic impurity in addition toadjustment of the ratios in volume of a plurality of regions relative tothe entire fractal structure. Of course, control of the quantum chaos inelectron states is possible merely by introducing a random magneticfield by addition of a magnetic impurity without adjustment of theratios in volume of a plurality of regions relative to the entirefractal structure. The ratios in volume of these regions correspond tothe ratios of numbers of atoms forming these regions, and in embodimentsof the invention explained later, they correspond to the ratios ofdurations of time for growth (step) required for forming these regions.

[0016] In the first aspect of the invention, the fractal structuretypically includes a first region forming the core having a firstfractal dimension, and one or more second regions surrounding the firstregion and having a second fractal dimension that is lower than thefirst fractal dimension. In the case where the entirety of the firstregion and the second region appears like a star, it is a stellarfractal structure. In the fractal structure having these first andsecond regions, from the viewpoint of ensuring sufficientcontrollability of the nature of phase transition or correlation betweeninteractive electrons in an electron system, that is, from the viewpointof forming a satisfactory junction between a Mott insulator and a metalby satisfactorily controlling the magnetization curve of ferromagneticphase transition, quantum chaos, and so on, or by satisfactorily formingD_(f1) and the second fractal dimension D_(f2) are determined to bepreferably D_(f1)>2.7 and D_(f2)<2.3, typically 2.7<D_(f1)≦3 and1≦D_(f2)<2.3, and more preferably 2.9≦D_(f1)≦3 and 1≦D_(f2)<2.3. Theupper limit value 3 of D_(f1) corresponds to the dimension of athree-dimensional space whereas the lower limit value 1 of D_(f2) isnecessary for ensuring connectivity in the structure.

[0017] According to the second aspect of the invention, there isprovided a method of forming a fractal structure having a plurality ofregions different in fractal dimension characterizing theself-similarity, comprising:

[0018] growing a fractal structure from one or more origins, andchanging growth conditions with time in the growth process thereof suchthat different fractal dimensions are obtained.

[0019] In the second aspect of the invention, there are used growthconditions ensuring the first fractal dimension to be made from thegrowth start point of time until a first point of time, and growthconditions ensuring a second fractal dimension lower than the firstfractal dimension to be made from the first point of time to a secondpoint of time. This results in forming a fractal structure including afirst region having the first fractal dimension and a second regionsurrounding the first region and having the second fractal dimensionlower than the first fractal dimension. In an embodiment that will beexplained later in detail, growth conditions of the fractal structureare represented by α of Equation (4). In actual growth, however, if thefractal structure is grown in the liquid phase, for example, natures ofsolvents employed for the growth are one of the growth conditions. Thatis, in this case, a plurality of regions different in fractal dimensioncan be formed by selecting appropriate solvents, respectively, in thegrowth process.

[0020] Similarly to the first aspect of the invention, in the fractalstructure including the first region and the second region, from theviewpoint of ensuring sufficient controllability of the nature of phasetransition or correlation between interactive electrons in an electronsystem, that is, from the viewpoint of forming a satisfactory junctionbetween a Mott insulator and a metal by satisfactorily controlling themagnetization curve of ferromagnetic phase transition, quantum chaos,and so on, or by satisfactorily forming Df, and the second fractaldimension D_(f2) are determined to be preferably D_(f1)>2.7 andD_(f2)<2.3, typically 2.7<D_(f1)<3 and 1≦D_(f2)<2.3, and more preferably2.9≦D_(f1)≦3 and 1≦D_(f2)<2.3. The upper limit value 3 of Df1corresponds to the dimension of a three-dimensional space whereas thelower limit value 1 of D_(f2) is necessary for ensuring connectivity inthe structure.

[0021] According to the invention having the above-summarizedconfiguration, a fractal structure, such as a stellar fractal structure,including a plurality of regions different in fractal dimension fromeach other can be obtained by changing the growth condition of thefractal structure with time. Then, in this fractal structure, the natureof phase transition occurring in the fractal structure can be controlledby adjusting the ratios in volume of the plurality of regions relativeto the entire fractal structure. Optimization of the fractal dimensionsalso improves the controllability.

BRIEF DESCRIPTION OF THE DRAWINGS

[0022]FIG. 1 is a schematic diagram that shows a stellar fractalstructure obtained by simulation according to the first embodiment ofthe invention;

[0023]FIG. 2 is a schematic diagram that shows a stellar fractalstructure obtained by simulation according to the first embodiment ofthe invention;

[0024]FIG. 3 is a schematic diagram for explaining dimensions of astellar fractal structure according to the first embodiment;

[0025]FIG. 4 is a schematic diagram that shows log-log plots of adistance r from the growth origin in the stellar fractal structureaccording to the first embodiment and the number of growth points N(r)contained in a ball with the radius r;

[0026]FIG. 5 is a schematic diagram that shows a relation between energyand density of states in the stellar fractal structure according to thefirst embodiment of the invention;

[0027]FIG. 6 is a schematic diagram that shows a relation between energyand density of states in the stellar fractal structure according to thefirst embodiment of the invention;

[0028]FIG. 7 is a schematic diagram that shows a relation between energyand density of states in the stellar fractal structure according to thefirst embodiment of the invention;

[0029]FIG. 8 is a schematic diagram for explaining spatial changesinside the stellar fractal structure according to the first embodimentof the invention;

[0030]FIG. 9 is a schematic diagram that shows log-log plots of adistance r from the growth origin in a stellar fractal structureaccording to the second embodiment and the number of growth points N(r)contained in a ball with the radius r;

[0031]FIG. 10 is a schematic diagram that shows log-log plots of adistance r from the growth origin in a stellar fractal structureaccording to the second embodiment and the number of growth points N(r)contained in a ball with the radius r;

[0032]FIG. 11 a schematic diagram that shows log-log plots of a distancer from the growth origin in a stellar fractal structure according to thesecond embodiment and the number of growth points N(r) contained in aball with the radius r;

[0033]FIG. 12 is a schematic diagram that shows spontaneousmagnetization in a fractal structure having a single fractal dimension;

[0034]FIG. 13 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the secondembodiment of the invention;

[0035]FIG. 14 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the secondembodiment of the invention;

[0036]FIG. 15 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the secondembodiment of the invention;

[0037]FIG. 16 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the third embodiment of theinvention;

[0038]FIG. 17 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the third embodiment of theinvention;

[0039]FIG. 18 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the third embodiment of theinvention;

[0040]FIG. 19 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the third embodiment of theinvention;

[0041]FIG. 20 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the third embodiment of theinvention;

[0042]FIG. 21 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the third embodiment of theinvention;

[0043]FIG. 22 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the fourth embodiment of theinvention;

[0044]FIG. 23 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the fourth embodiment of theinvention;

[0045]FIG. 24 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the fourth embodiment of theinvention;

[0046]FIG. 25 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the fourth embodiment of theinvention;

[0047]FIG. 26 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the fourth embodiment of theinvention;

[0048]FIG. 27 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the fourth embodiment of theinvention;

[0049]FIG. 28 is a schematic diagram that shows a relation between a andthe fractal dimension D_(f) in a simplex fractal structure;

[0050]FIG. 29 is a schematic diagram that shows M_(s)(r) in a stellarfractal structure according to the fifth embodiment of the invention;

[0051]FIG. 30 is a schematic diagram that shows M_(s)(r) in a stellarfractal structure according to the fifth embodiment of the invention;

[0052]FIG. 31 is a schematic diagram that shows M_(s)(r) in a stellarfractal structure according to the fifth embodiment of the invention;

[0053]FIG. 32 is a schematic diagram that shows D_(ave)(r) in a stellarfractal structure according to the fifth embodiment of the invention;

[0054]FIG. 33 is a schematic diagram that shows D_(ave)(r) in a stellarfractal structure according to the fifth embodiment of the invention;

[0055]FIG. 34 is a schematic diagram that shows D_(ave)(r) in a stellarfractal structure according to the fifth embodiment of the invention;

[0056]FIG. 35 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the sixthembodiment of the invention;

[0057]FIG. 36 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the sixthembodiment of the invention;

[0058]FIG. 37 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the sixthembodiment of the invention;

[0059]FIG. 38 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the sixthembodiment of the invention;

[0060]FIG. 39 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the sixthembodiment of the invention;

[0061]FIG. 40 is a schematic diagram that shows spontaneousmagnetization in a stellar fractal structure according to the sixthembodiment of the invention;

[0062]FIG. 41 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention;

[0063]FIG. 42 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention;

[0064]FIG. 43 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention;

[0065]FIG. 44 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention;

[0066]FIG. 45 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention;

[0067]FIG. 46 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention;

[0068]FIG. 47 is a schematic diagram that shows quantum level statisticsin-a stellar fractal structure according to the seventh embodiment ofthe invention;

[0069]FIG. 48 is a schematic diagram that shows quantum level statisticsin a stellar fractal structure according to the seventh embodiment ofthe invention; and

[0070]FIG. 49 is a schematic diagram that shows the Berry-Robnikparameter ρ in a star-shaped fractal structure according to the seventhembodiment of the invention.

BEST MODE FOR CARRYING OUT THE INVENTION

[0071] Some embodiments of the invention will now be explained below. Inthe following embodiments, stellar fractal structures are taken as onetype of fractal structures including a plurality of portionscharacterized in a plurality of fractal dimensions. These stellarfractal structures each have a form of a nerve cell obtained by forminga region with a low fractal dimension around a core with a high fractaldimension in the process of growing random fractals.

[0072] First Embodiment

[0073] (1) Formation of a Stellar Fractal Structure

[0074] A method of forming a stellar fractal structure according to thefirst embodiment can be obtained by developing Dielectric BreakdownModel ((32) A. Erzan, L. Pietronero, A. Vespignani, Rev. Mod. Phys. 67,545(1995); (33) L. Niemeyer, L. Pietronero, H. J. Wiesmann, Phys. Rev.Lett. 52, 1033(1984)).

[0075] Here is defined a cubic lattice S in a three-dimensional space,and a scalar potential field φ(i₁, i₂, i₃) is defined on its latticesite (i₁, i_(2,) i₃)∈S and called a potential. Let this potential obeythe Laplace's equation

Δφ(i ₁ ,i ₂ ,i ₃)=0  (1)

[0076] A pattern T_(n), which will be defined later, is a set of latticesites on the three-dimensional lattice. T₀ contains (0, 0, 0) alone, andT_(n+1) is created by sequentially adding a single lattice site to T_(n)according to the rule given below.

[0077] Let the potential of each site contained in T_(n) be I and letthe potential at infinity be 0. That is,

φ(i₁ ,i ₂ ,i ₃)=0 when (i₁ ,i ₂ ,i ₃)→∞  (2)

φ(i₁ ,i ₂ ,i ₃)=1 when (i₁ ,i ₂ ,i ₃)∈T _(n)  (3)

[0078] Under these boundary conditions, Equation (1) is solved todetermine the potential. The lattice site, which will be added to T_(n)to construct T_(n+1), is not contained in T_(n), and it is selected fromthe set of lattice sites, i.e. U_(n), nearest to T_(n). The number oflattice sites contained in U_(n) is denoted by N_(n).

[0079] Intensity of the electric fields for lattice sites (i_(1,m),i_(2,m), i_(3,m)) (where m=1, 2, . . . , N_(n)) in U_(n) is defined as

E_(m)(α)=|φ(i _(1,m) ,i _(2,m) ,i _(3,m))−1|^(α)  (4)

[0080] The probability that a certain site (i_(1,m), i_(2,m), i_(3,m))in U_(n) is selected is proportional to the electric field E_(m)(α).That is, the probability is $\begin{matrix}{{p_{m}(\alpha)} = \frac{E_{m}(\alpha)}{\sum\limits_{j = 1}^{N_{n}}{E_{j}(\alpha)}}} & (5)\end{matrix}$

[0081] By repeating these operations, construction of T_(n) isprogressed. An ideal fractal will be a set of r limits repeatedinfinitely as $\begin{matrix}{T_{\infty} = {\underset{narrow\infty}{\lim \quad}T_{n}}} & (6)\end{matrix}$

[0082] When α=1, the foregoing and the result of generation of thepattern by Diffusion limited aggregation are consistent ((34) T. A.Witten, Jr. and L. M. Sander, Phys. Rev. Lett. 47, 1400(1984); Phys.Rev. B 27, 5686(1983)).

[0083] The stellar fractal structure according to the first embodimentis formed by changing the a parameter in accordance with the step n ofthe above-mentioned growth. That is, the above growth process isprogressed using α₃ under 1≦n≦τ₁, α₂ under τ₁+1≦n≦τ₂, and α₃ underτ₂+1≦n≦Σ₃. Simulation is performed below, taking fractal structurehaving two different fractal dimensions. Results of the simulation areshown in FIGS. 1, 2 and 3. In that simulation, T₂=10000, α₁=0 and α₂=2are commonly used in all cases whereas τ₁=3000 is used in FIG. 1,T₁=5000 is used in FIG. 2 and τ₁=7000 is used in FIG. 3, respectively.It will be understood from FIGS. 1, 2 and 3 that the fractal structureincludes a ball-shaped region with a high fractal dimension (somaticfractal) at the center (growth origin) and a tree-like region with a lowfractal dimension (dendritic fractal) around it. Nerve cells oftenexhibit this type of structure. That is, tree-like projections extendfrom the soma of a cell and perform their function.

[0084] For the purpose of understanding the structure in greater detail,a calculation process of fractal dimensions is used. Let r represent thedistance from the growth origin (0, 0, 0) and N(r) represent the numberof growth points contained in a ball with the radius r. Then, using a asa factor of proportionality, if N(r) can be expressed as

N(r)=ar ^(D) ^(_(f))   (7)

[0085] then D_(f) is called the fractal dimension. Therefore, sincelogarithms of both sides are

log N(r)=log a+D _(f) logr  (8)

[0086] if log-log plots ride on a straight line, the structure can beregarded as a fractal, and its inclination is the fractal dimension. InFIG. 4, log-log plotting was made for various cases obtained by theabove-mentioned growth experiment. It is found from FIG. 4 that, under asmall log(r), plots ride on a straight line with a large inclination,but they ride on a straight line with a small inclination when log(r)exceeds a certain critical point. This demonstrates that the structureinclude two different fractal dimensions. More specifically, in thisexample, the fractal dimension before log(r) exceeds the critical pointis approximately 2.9, and excluding the case of τ₁=8000, the fractaldimension after log(r) exceeds the critical point is approximately 2.1.Sites of the critical point increase as τ₁ increases, and this supportsthis growth model and its interpretation.

[0087] (2) Correlative Electron System in a Stellar Fractal Structure

[0088] An electron system is defined on the stellar fractal structuredefined in (1). A review is made about the lattice point

r _(p)=(i _(1,p) i _(2,p) ,i _(3,p))∈T_(n)  (9)

[0089] that is the origin of T_(n). In Equation (9), p=1, 2, . . . ,n+1. An operator ĉ_(p,σ) ^(†) for generating an electron of a spin σ isdefined at the lattice point r_(p)∈T_(n). Of course, there is theanticommutative relation

{ĉ_(p,σ),ĉ_(q,ρ) ^(†)}=δ_(p,q)δ_(σ,ρ)  (10)

[0090] Here is defined a single-band Hubbard Hamiltonian Ĥ of theelectron system as follows. $\begin{matrix}{\hat{H} = {{t{\sum\limits_{i,j,\sigma}{\lambda_{i,j}{\hat{c}}_{i,\sigma}^{\dagger}{\hat{c}}_{j,\sigma}}}} + {U{\sum\limits_{j}{{\hat{n}}_{j, \uparrow}{\hat{n}}_{j, \downarrow}}}}}} & (11)\end{matrix}$

[0091] Assuming that electrons are movable only among nearest-neighborsites, the following is employed as λ_(p,q). $\begin{matrix}{\lambda_{p,q} = \{ \begin{matrix}1 & {\quad { {when}\quad \middle| {r_{p} - r_{q}} | = 1}} \\0 & {\quad {otherwise}}\end{matrix} } & (12)\end{matrix}$

[0092] Additionally, the spin σ electron density operator of the j-thsite, {circumflex over (n)}_(j,σ)=ĉ_(j,σ) ^(↑)ĉ_(j,σ), and their sum,{circumflex over (n)}_(j)=Σ_(σ){circumflex over (n)}_(j,σ),_(σ){circumflex over (n)}_(j,σ), are defined.

[0093] For the purpose of defining a temperature Green's function, hereis introduced a grand canonical Hamiltonian {circumflex over(K)}=Ĥ−μ{circumflex over (N)} where {circumflex over(N)}=Σ_(j){circumflex over (n)}_(j). In the half-filled taken here,chemical potential is μ=U/2. The half-filled grand canonical Hamiltoniancan be expressed as $\begin{matrix}{\hat{K} = {{t{\sum\limits_{i,j,\sigma}{\lambda_{j,i}{\hat{t}}_{j,i,\sigma}}}} + {{U/2}{\sum\limits_{i}( {{\hat{u}}_{i} - 1} )}}}} & (13)\end{matrix}$

[0094] Operators {circumflex over (t)}_(j,i,Σ), ĵ_(j,i,σ), û_(i) and{circumflex over (d)}_(i,σ) are defined beforehand as

{circumflex over (t)} _(j,i,σ) =ĉ _(j,σ) ^(†) ĉ _(i,σ) +ĉ _(i,σ) ^(†) ĉ_(j,σ)  (14)

ĵ _(j,i,σ) =ĉ _(j,σ) ^(†) ĉ _(i,σ) −ĉ _(i,σ) ^(†) ĉ _(j,σ)  (15)

û _(i) =ĉ _(i,↑) ^(†) ĉ _(i,↑) ĉ _(i,↓) ^(†) ĉ _(i,↓) +ĉ _(i,↑) ĉ _(i,↑)^(†) ĉ _(i,↓) ĉ _(i,↓) ^(†)  (16)

{circumflex over (d)} _(i,σ) =ĉ _(i,σ) ^(†) ĉ _(i,σ) −ĉ _(i,σ) ĉ _(i,σ)^(†)  (17)

[0095] If the temperature Green function is defined for operators Â and{circumflex over (B)} given, taking τ as imaginary time, it is asfollows.

<Â;{circumflex over (B)}>=−∫ ₀ ^(β) dτ<T _(τ) Â(τ){circumflex over(B)}>e ^(iω) ^(_(n)) ^(τ)  (18)

[0096] The on-site Green function

G _(j,σ)(iω _(n))=<ĉ _(j,σ) ;ĉ _(j,σ) ^(†)>  (19)

[0097] is especially important.

[0098] Imaginary time development of the system is obtained by theHeisenberg equation $\begin{matrix}{{\frac{\quad}{\tau}{\hat{A}(\tau)}} = \lbrack {\hat{K},\hat{A}} \rbrack} & (20)\end{matrix}$

[0099] As the equation of motion of the on-site Green function,$\begin{matrix}{{i\quad \omega_{n}{\langle{{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}} = {1 + {t{\sum\limits_{p,j}{\lambda_{p,j}{\langle{{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}} + {\frac{U}{2}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}} & (21)\end{matrix}$

[0100] is obtained. Then, the approximation shown below is introduced,following Gros ((35) C. Gros, Phys. Rev. B50, 7295(1994)). If the site pis the nearest-neighbor site of the site j, the resolution

<ĉ _(i,σ) ;ĉ _(j,σ) ^(†) >→t<ĉ _(p,σ) ;ĉ _(p,σ) ^(†) ><ĉ _(j,σ) ;ĉ_(j,σ) ^(†)>  (22)

[0101] is introduced as the approximation. This is said to be exact incase of infinite-dimensional Bethe lattices, but in this case, it isonly within approximation. Under the approximation, the followingequation is obtained. $\begin{matrix}{{{( {{i\quad \omega_{n}} - {t^{2}\Gamma_{j,\sigma}}} )G_{j,\sigma}} = {1 + {\frac{U}{2}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}}{where}} & (23) \\{\Gamma_{j,\sigma} = {\sum\limits_{p}{\lambda_{p,j}G_{p,\sigma}}}} & (24)\end{matrix}$

[0102] was introduced. To solve the equation, obtained, <{circumflexover (d)}_(j,−σ)ĉ_(j,σ);ĉ_(j,σ) ^(†)> has to be analyzed. In case ofhalf-filled models, this equation of motion is $\begin{matrix}{{i\quad \omega_{n}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}} = {{\frac{U}{2}G_{j,\sigma}} - {2t{\sum\limits_{p}{\lambda_{p,j}{\langle{{{\hat{j}}_{p,j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}} + {t{\sum\limits_{p}{\lambda_{p,j}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{p,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}}}} & (25)\end{matrix}$

[0103] Here again, with reference to the Gros theory, approximation isintroduced. It is the following translation.

<ĵ _(p,j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†) >→tG _(p,−σ) <{circumflex over (d)}_(j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†>)  (26)

<{circumflex over (d)} _(j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†) >→tG _(p,σ)<{circumflex over (d)} _(j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†)>  (27)

[0104] By executing this translation, the following closed equation isobtained. $\begin{matrix}{{( {{i\quad \omega_{n}} - {t^{2}\Gamma_{j,\sigma}}} )G_{j,\sigma}} = {1 + {\frac{( {U/2} )^{2}}{{i\quad \omega_{n}} - {t^{2}\Gamma_{j,\sigma}} - {2t^{2}\Gamma_{j,{- \sigma}}}}G_{j,\sigma}}}} & (28)\end{matrix}$

[0105] Here is assumed that there is no dependency on spin. That is,assuming

G _(j,↑) =G _(j,↓)  (29)

[0106] the following calculation is executed. This is because, whenanalytic continuation is conducted as iω_(n)→ω+iδ for small δ,

D _(j)(ω)=−ImG _(j)(ω+iδ)  (30)

[0107] becomes the local density of states of the site j, and$\begin{matrix}{{D(\omega)} = {{- \frac{1}{n + 1}}{\sum\limits_{j}{D_{j}(\omega)}}}} & (31)\end{matrix}$

[0108] becomes the density of states of the system. For later numericalcalculation of densities of states, δ=0.0001 will be used.

[0109] Regarding the stellar fractal structure obtained in (1), its Motttransition is analyzed. There were commonly used τ₂=10000, α₁=0 andα₂=2, and used as τ₁ were τ₁=0, 2000, 4000, 6000, 8000 and 10000.Densities of states obtained by numerical calculation using τ=1 areshown in FIGS. 5, 6 and 7. U=8 was used in FIG. 5, U=10 in FIG. 6, andU=12 in FIG. 7. In FIG. 5, in case of τ₁=10000, that is, in case of apure fractal structure of α=0, the density of states D(0) under ω=0 isfinite, and this electron system is in the metal phase. As τ₁ becomessmaller, D(0) gradually decreases, and the nature as an insulator isenhanced. Under τ₁<4000, almost all of D(0) disappears, and Mottinsulator transformation occurs.

[0110] In FIG. 6, the system behaves as a Mott insulator under allparameters; however, its insulating performance, i.e.electron-to-electron correlation effect, increases as τ_(i) varies. Thisis because the effective Hubbard bandwidth increases along with adecrease of τ₁.

[0111] Spatial changes inside a stellar fractal structure are reviewedbelow in detail. For this purpose, spatial changes of the local densityof states D_(j) (ω=0) on the Fermi energy are analyzed. When thedistance from the growth origin (0, 0, 0) is r, the set of growth pointsdistant from the growth origin by r to r+dr is written as U_(s)(r), andits original number is written as M_(s)(r). Then, D_(ave)(r) is definedas follows. $\begin{matrix}{{D_{ave}(r)} = {\frac{1}{M_{s}(r)}{\sum\limits_{j \in {U_{s}{(r)}}}{D_{j}(0)}}}} & (32)\end{matrix}$

[0112] This amount is the average value of local densities of states onthe Fermi energy on the spherical surface with the distance r from thegrowth origin. In FIG. 8, values of D_(ave)(r) under U=8 are shown. Inthis calculation, dr=1 was used, and values of D_(ave)(r) were plottedfor various values of τ₁. In the region with small r, that is, in theregion of the somatic fractal region, D_(ave)(r) takes finite values,and the system behaves as a metal. As r increases, D_(ave)(r) decreasesand approaches zero asymptotically. That is, Mott insulatortransformation occurs. The value of r causing Mott insulatortransformation varies, and it will be needless to say that the larger τ₁the larger value of r is required, considering that Mott metal-insulatortransition can be controlled with the α parameter. That is, it resultsin realizing a new electron system in which the central metal region iswrapped by the Mott insulator region.

[0113] Thus, it has been confirmed that, in the star-shaped fractalstructure, the Mott metal-insulator transition occurring in thestructure and the nature of the Mott insulator can be controlled with l.

[0114] Second Embodiment

[0115] (1) Formation of a Stellar Fractal Structure

[0116] A method of forming the star-shaped fractal structure accordingto the second embodiment is the same as the method of forming thestellar fractal structure according to the first embodiment. As growthconditions, however, (α₁, α₂)=(0, 1) and (α₁, α₂)=(1, 2) were used inaddition to (α₁, α₂)=(0, 2), and numerical experiment of growth wasconducted for these cases.

[0117] In FIGS. 9 through 11, for respective examples obtained by thegrowth experiment, log-log plotting was made similarly to the firstembodiment. (α₁, α₂)=(0, 2) was used in FIG. 9, (α₁, α2)=(0, 1) was usedin FIG. 10, and (α₁, α₂)=(1, 2) was used in FIG. 11. As shown in FIGS. 9through 11, when log(r) is small, plots ride on a straight line with alarge inclination. When log(r) exceeds a certain critical point, plotsride on a straight line with a small inclination. This means that thestructure includes two different fractal dimensions. The site of thecritical point increases as τ₁ increases, and this supports this growthmodel and its interpretation.

[0118] (2) Ferromagnetic Transition in a Stellar Fractal Structure

[0119] A spin system describing a ferromagnetic structure is defined onthe stellar fractal structure defined in (1). A review is made about thelattice point

r _(p)=(_(1,p) _(2,p) ,i _(3,p))∈T_(n)  (33)

[0120] that is the origin of T_(n). In Equation 33, p=0, 1, 2, . . . ,n. By placing a spin on a lattice site r_(p)∈T_(n), here is taken a spinsystem that can be described by the following Hamiltonian.$\begin{matrix}{H = {- {\sum\limits_{p,q}{J_{p,q}{S_{p} \cdot S_{q}}}}}} & (34)\end{matrix}$

[0121] S_(p) is the spin at the p site. A natural model of spin-spininteraction J_(p,q) is: $\begin{matrix}{J_{p,q} = \{ \begin{matrix}1 & {\quad { {when}\quad \middle| {r_{p} - r_{q}} | = 1}} \\0 & {\quad {otherwise}}\end{matrix} } & (35)\end{matrix}$

[0122] That is, spin-spin interaction exists only betweennearest-neighbor sites. For the purpose of calculating spontaneousmagnetization M at a finite temperature T, statistical mechanics of anequilibrium system is introduced. A partition function Z is defined asfollows. $\begin{matrix}{Z = {\sum\limits_{\{ s_{p}\}}^{{- H}/T}}} & (36)\end{matrix}$

[0123] where {S_(p)} in the symbol of the sum pertains to the sumregarding all spin states. Spontaneous magnetization is defined as thestatistical average of spins as follows. $\begin{matrix}{M = {\frac{1}{n + 1}{\sum\limits_{p = 1}^{n + 1}{\langle S_{p}\rangle}}}} & (37)\end{matrix}$

[0124] where the expected value <S_(p)> is $\begin{matrix}{{\langle S_{p}\rangle} = {\frac{1}{Z}{\sum\limits_{\{ s_{p}\}}{S_{p}^{{- H}/T}}}}} & (38)\end{matrix}$

[0125] and n+1 is the total number of spins. M is generally a vectorquantity in a spin space, but its absolute value M=|M| is calculated.

[0126] Here is made a review about an Ising model. In an Ising model,only two states of

S _(P)=1 or −1  (39)

[0127] can exist. Let a mean field approximation be introduced into theIsing model. Spontaneous magnetization of the p-th site is written asμ_(p). In flank this system, since the molecule field varies with site,let it written as {overscore (μ)}_(p). As an assumption of the meanfield approximation, here is employed a molecule field that can bewritten by spontaneous magnetization of the nearest-neighbor site as$\begin{matrix}{{\overset{\_}{\mu}}_{p} = {\sum\limits_{q}{J_{p,q}\mu_{q}}}} & (40)\end{matrix}$

[0128] This assumption simplifies the foregoing Hamiltonian to$\begin{matrix}{H_{M\quad F} = {- {\sum\limits_{P = 1}^{n + 1}{{\overset{\_}{\mu}}_{p}\sigma_{p}}}}} & (41)\end{matrix}$

[0129] A self-consistent equation ensuring spontaneous magnetizationobtained by using a partition function by the simplified Hamiltonianbecomes μ_(p) results in

μ_(p)=tan h(β{overscore (μ)}_(p))  (42)

[0130] and by numerically solving this equation, spontaneousmagnetization of the system $\begin{matrix}{M_{I\quad \sin \quad g} = {\frac{1}{n + 1}{\sum\limits_{j = 0}^{n}\mu_{j}}}} & (43)\end{matrix}$

[0131] is obtained.

[0132]FIG. 12 shows spontaneous magnetization in a typical fractalstructure having a single fractal w dimension. Here are shown results ofα=0, α=1 and α=2. Now, spontaneous magnetization in a star-shapedfractal structure is demonstrated. FIGS. 13 through 15 spontaneousmagnetization in stellar fractal structures, for (α₁, α₂)=(0, 2) in FIG.13, for (α₁, C₂)=(0, 1) in FIG. 14, and (α₁, α₂)=(1, 2) in FIG. 15. FIG.13 is discussed in greater detail. In case of τ₁=0, spontaneousmagnetization is the same as that of a simple fractal structure of α=2.On the other hand, in case of τ₁=10000, it is the same as that of asimple fractal structure of α=0. When intermediate values between thosetwo poles are taken sequentially, a new magnetization curveinterpolating the two poles appears. Of course, this magnetization curveis near to the weighted mean of individual portions of the two poles.Actually, however, there is an interaction between the region of α=2 andthe region of α=0, and the structure construct a new magnetic structureas a whole. Also in FIG. 14, a new magnetization curve interpolatingsimple fractal structures of α=1 and α=0 appears, and in FIG. 15, a newmagnetization curve interpolating simple fractal structures of α=2 andα=1 appears. It has been found, therefore, that the use of this stellarfractal structure, which is a fractal-based complex, leads torealization of materials exhibiting various magnetic properties.

[0133] Third Embodiment

[0134] (1) Formation of a Stellar Fractal Structure

[0135] A method of forming the star-shaped fractal structure accordingto the third embodiment is the same as the method of forming the stellarfractal structure according to the first embodiment. As growthconditions, however, similarly to the second embodiment, (α₁, α₂)=(0, 1)and (α₁, α₂)=(1, 2) were used in addition to (α₁, α₂)=(0, 2), andnumerical experiment of growth was conducted for these cases. Results oflog-log plotting for respective cases obtained by the growth experimentare the same as those shown in FIGS. 9 through 11.

[0136] (2) Electron System on a Stellar Fractal Structure

[0137] Let us define a quantum system of one particle on the stellarfractal defined in (1). Assume a lattice site shown below, which is theorigin of T_(n).

r _(p)=(i _(1,p) ,i _(2,p) ,i _(3,p))∈T_(n)  (44)

[0138] where p=0, 1, 2, . . . , n. Here is defined an operator ĉ_(p)^(†) that creates a quantum at a lattice site r_(p)∈T_(n). Of course, ananticommutative relation

{ĉ _(p) ,ĉ _(q) ^(†)}=δ_(p,q)  (45)

[0139] is established. Here the Hamiltonian Ĥ is defined as$\begin{matrix}{\hat{H} = {- {\sum\limits_{p,q}\quad {t_{p,q}{\hat{c}}_{p}^{\dagger}{\hat{c}}_{q}}}}} & (46)\end{matrix}$

[0140] Here is employed as the transfer t_(p,q), $\begin{matrix}{t_{p,q} = \{ \begin{matrix}1 & {{{when}\quad {{r_{p} - r_{q}}}} = 1} \\0 & {otherwise}\end{matrix} } & (47)\end{matrix}$

[0141] In this model, hopping is possible only between nearest-neighborsites.

[0142] When ε_(m) denotes the eigenenergy of the Hamiltonian Ĥ and |m>denotes the eigenvector,

Ĥ|m>=ε _(m) m>  (48)

[0143] where m=0, 1, 2, . . . , n.

[0144] First, n+1 quantum levels ε_(m) are standardized such thatspacing between nearest-neighbor levels becomes 1 in average. That is,

ω_(j)=ε−ε_(j−1)  (49)

[0145] However, when j=1, 2, . . . , n, by using $\begin{matrix}{\overset{\_}{\omega} = {\frac{1}{n}{\sum\limits_{j = 1}^{n}\quad \omega_{j}}}} & (50)\end{matrix}$

[0146] it is converted into new levels

ε₀=0  (51)

[0147] $\begin{matrix}{{ɛ_{m} = {{\frac{1}{\overset{\_}{\omega}}{\sum\limits_{j = 1}^{m}\quad \omega_{j}}} = {\sum\limits_{j = 1}^{m}\quad \Omega_{j}}}}{{Here},}} & (52) \\{\Omega_{j} = \frac{\omega_{j}}{\overset{\_}{\omega}}} & (53)\end{matrix}$

[0148] The density of states of the system is defined by $\begin{matrix}{{p(ɛ)} = {\frac{1}{n + 1}{\sum\limits_{m = 1}^{n + 1}\quad {\delta ( {ɛ - ɛ_{m}} )}}}} & (54)\end{matrix}$

[0149] and the staircase function

λ(ε)=∫_(−∞) ^(ε) dηρ(η)  (54)

[0150] is calculated. The staircase function obtained is converted by atechnique called “unfolding” such that the density of states becomesconstant in average. By using quantum levels obtained in this manner,nearest-neighbor level spacing distribution P(s) and Δ₃ statistics ofDyson and Metha are calculated as quantum level statistics. As taught ina literature ((36) L. E. Reichl, The transition to chaos: inconservative classical systems: quantum manifestations (Springer, NewYork, 1992) (37) F. Haake, Quantum Signatures of chaos,(Springer-Verlag, 1991)), by using these statistics, it can be detectedwhether quantum chaos has been generated or not. It is also known that aquantum chaotic system is sensitive to perturbation from outsidesimilarly to the classical chaotic system, and analysis of quantum chaosis important as a polestar of designs of non-linear materials.

[0151] In case of an integrable system, nearest-neighbor level spacingdistribution P(s) and Δ₃ statistics are those of Poisson distribution

P _(P)(s)=e ^(−S)  (56)

[0152] $\begin{matrix}{{\Delta_{3}(n)} = \frac{n}{15}} & (57)\end{matrix}$

[0153] In case of a quantum chaotic system, it becomes GOE distribution$\begin{matrix}{{P_{GOE}(s)} = {\frac{\pi \quad s}{2}e^{{- \pi}\quad {s^{2}/4}}}} & (58) \\{{\Delta_{3}(n)} = {{\frac{1}{\pi^{2}}\lbrack {{\log ( {2\pi \quad n} )} + \gamma - \frac{\pi^{2}}{8} - \frac{5}{4}} \rbrack} + {O( n^{- 1} )}}} & (59)\end{matrix}$

[0154] where γ is the Euler's constant.

[0155] Since the stellar fractal structure analyzed here is one obtainedby growth experiment of n=10000, this quantum system includes n+1=10001eigenstates. Based on energy eigenvalues concerning 1501 states from the510-th to 2001-th eigenstates from the ground state, among thoseeigenstates, the following quantum level statistics was calculated.FIGS. 16 and 17 show quantum level statistics in a stellar fractalstructure of (α₁, α₂)=(0, 2). FIG. 16 shows P(s) and FIG. 17 shows Δ₃statistics. In case of τ₁=10000, the structure is the same as the simplefractal of α=0, and the fractal dimension is near 3 (D_(f)˜2.91).Therefore, the system behaves as a quantum chaotic system. In this case,the quantum level statistics is that of GOE distribution. On the otherhand, in case of τ₁=0, the structure is the same as the simple fractalof α=2, and the fractal dimension is near 2 (D_(f)˜2.16). Therefore, thesystem behaves as an integrable system. Then as τ₁ increases from 0 to10000, the quantum level statistics changes from Poisson distribution toGOE distribution. Therefore, wide kinds of quantum systems can berealized by setting τ₁ at predetermined values.

[0156] The above examples are stellar fractal structures in which α=0and α=2 coexist. In contrast, a simple fractal structure of α=1 is of alevel of the fractal dimension (D_(f)˜2.45), and a quantum chaoticsystem of α=0 and an integrable system can be connected via the simplefractal of α=1. FIGS. 18 and 19 show quantum level statistics in astellar fractal structure of (α₁, α₂)=(0, 1). FIGS. 20 and 21 showquantum level statistics in a stellar fractal structure of (α₁, α₂)=(1,2). FIGS. 18 and 20 show P(s), and FIGS. 19 and 21 show Δ₃ statistics.τ₁=10000 of FIGS. 18 and 19 is the same quantum chaotic system asτ₁=10000 of FIGS. 16 and 17. As shown in FIGS. 18 and 19, as τ₁decreases, the quantum level statistics changes from that of GOEdistribution to that of Poisson distribution. However, even in τ₁=0 ofFIGS. 18 and 19, i.e., even in the simple fractal of α=1, it does notexhibit complete Poisson distribution. Let us change the eyes to FIGS.20 and 21. τ₁=10000 of FIGS. 20 and 21 is the same as τ₁=0 of FIGS. 18and 19, and it is the quantum level statistics in the simple fractalstructure of α=1. As τ₁ decreases, the quantum level statisticsapproaches that of Poisson distribution, and in a region with small τ₁,it asymptotically approaches that of Poisson distribution. Of course,τ₁=0 of FIGS. 20 and 21 is the quantum level statistics in the simplefractal of α=2, and it is the same as τ₁=0 of FIGS. 16 and 17.

[0157] Fourth Embodiment

[0158] (1) Formation of a Stellar Fractal Structure

[0159] A method of forming the star-shaped fractal structure accordingto the fourth embodiment is the same as the method of forming thestellar fractal structure according to the first embodiment. As growthconditions, however, similarly to the second embodiment, (α₁, α₂)=(0, 1)and (α₁, α₂)=(1, 2) were used in addition to (α₁, α₂)=(0, 2), andnumerical experiment of growth was conducted for these cases. Results oflog-log plotting for respective cases obtained by the growth experimentare the same as those shown in FIGS. 9 through 11.

[0160] (2) Electron System on a Stellar Fractal Structure

[0161] Let us define a quantum system of one particle on the stellarfractal defined in (1). Assume a lattice site shown below, which is theorigin of T_(n).

r _(p)=(i _(1,p) ,i _(2,p) ,i _(3,p))∈T _(n)  (60)

[0162] where p=0, 1, 2, . . . , n. Here is defined an operator ĉ_(p)^(†) that creates a quantum at a lattice site r_(p)∈T_(n). Of course, ananticommutative relation

{ĉ _(p) ,ĉ _(q) ^(†)}=δ_(p,q)  (61)

[0163] is established. Here the Hamiltonian Ĥ is defined as$\begin{matrix}{\hat{H} = {- {\sum\limits_{p,q}\quad {t_{p,q}{\hat{c}}_{p}^{\dagger}{\hat{c}}_{q}}}}} & (62)\end{matrix}$

[0164] Here is employed as the transfer t_(p,q), $\begin{matrix}{t_{p,q} = \{ \begin{matrix}{\exp ( {\theta}_{p,q} )} & {{{when}\quad {{r_{p} - r_{q}}}} = 1} \\0 & {otherwise}\end{matrix} } & (63)\end{matrix}$

[0165] In this equation, θ_(p,q)=−θ_(q,p) is a random real numbersatisfying

0<θ_(p,q)<2π  (64)

[0166] In this model, hopping is possible only between nearest-neighborsites. Then, along with the hopping, rE phase factors θ_(p,q), which arerandom from one site to another, are added. If the phase factor isintegrated in the loop making one turn around a lattice point, amagnetic flux passing through the loop is obtained. This means amagnetic field is locally introduced to the random distribution of0<θ_(p,q)<2π. This magnetic field is absolutely random in both intensityand direction, and in spatial average, it becomes a zero magnetic fieldand never breaks the fractal property of the system.

[0167] When ε_(m) denotes the eigenenergy of the Hamiltonian Ĥ and |m>denotes the eigenvector,

Ĥ|m>=ε _(m) |m>  (65)

[0168] where m=0, 1, 2, . . . , n.

[0169] First, n+1 quantum levels ε_(m) are standardized such thatspacing between nearest-neighbor levels becomes 1 in average. That is,

ω_(j)=ε_(j)−ε_(j−1)  (66)

[0170] However, when j=1, 2, . . . , n, by using $\begin{matrix}{\overset{\_}{\omega} = {\frac{1}{n}{\sum\limits_{j = 1}^{n}\quad \omega_{j}}}} & (67)\end{matrix}$

[0171] it is converted into new levels

ε₀=0  (68)

[0172] $\begin{matrix}{{ɛ_{m} = {{\frac{1}{\overset{\_}{\omega}}{\sum\limits_{j = 1}^{n}\omega_{j}}} = {\sum\limits_{j = 1}^{m}\Omega_{j}}}}{{Here},}} & (69) \\{\Omega_{j} = \frac{\omega_{j}}{\overset{\_}{\omega}}} & (70)\end{matrix}$

[0173] The density of states of the system is defined by $\begin{matrix}{{p(ɛ)} = {\frac{1}{n + 1}{\sum\limits_{m = 1}^{n + 1}{\delta ( {ɛ - ɛ_{m}} )}}}} & (71)\end{matrix}$

[0174] and the staircase function

λ(ε)=∫_(−∞) ^(ε) dηρ(η)  (72)

[0175] is calculated. The staircase function obtained is converted by atechnique called “unfolding” such that the density of states becomesconstant in average. By using quantum levels obtained in this manner,nearest-neighbor level spacing distribution P(s) and Δ₃ statistics ofDyson and Metha are calculated as quantum level statistics. As taught ina literature ((36) L. E. Reichl, The transition to chaos: inconservative classical systems: quantum manifestations (Springer, NewYork, 1992) (37) F. Haake, Quantum Signatures of chaos,(Springer-Verlag, 1991)), by using these statistics, it can be detectedwhether quantum chaos has been generated or not. It is also known that aquantum chaotic system is sensitive to perturbation from outsidesimilarly to the classical chaotic system, and analysis of quantum chaosis important as a polestar of designs of non-linear materials.

[0176] In case of an integrable system, nearest-neighbor level spacingdistribution P(s) and Δ₃ statistics are those of Poisson distribution

P _(P)(s)=e ^(−S)  (73)

[0177] $\begin{matrix}{{\Delta_{3}(n)} = \frac{n}{15}} & (74)\end{matrix}$

[0178] In case of a quantum chaotic system under a magnetic field, itbecomes GUE (Gaussian unitary ensemble) distribution $\begin{matrix}{{P_{GUE}(s)} = {\frac{32s^{2}}{\pi^{2}}^{{- 4}{s^{2}/\pi}}}} & (75) \\{{\Delta_{3}(n)} = {{\frac{1}{2\pi^{2}}\lbrack {{\log ( {2\pi \quad n} )} + \gamma - \frac{5}{4}} \rbrack} + {O( n^{- 1} )}}} & (76)\end{matrix}$

[0179] where γ is the Euler's constant.

[0180] Since the stellar fractal structure analyzed here is one obtainedby growth experiment of n=10000, this quantum system includes n+1=10001eigenstates. Based on energy eigenvalues concerning 1501 states from the510-th to 2001-th eigenstates from the ground state, among thoseeigenstates, the following quantum level statistics was calculated.FIGS. 22 and 23 show quantum level statistics in a stellar fractalstructure of (α₁, α₂)=(0, 2). FIG. 22 shows P(s) and FIG. 23 shows Δ₃statistics. In case of τ₁=10000, the structure is the same as the simplefractal of α=0, and the fractal dimension is near 3 (D_(f)˜2.91).Therefore, the system behaves as a quantum chaotic system. In this case,the quantum level statistics is that of GOE distribution. On the otherhand, in case of T₁=0, the structure is the same as the simple fractalof α=2, and the fractal dimension is near 2 (D_(f)˜2.16). Therefore, thesystem behaves as an integrable system. Then as τ₁ increases from 0 to10000, the quantum level statistics changes from Poisson distribution toGOE distribution. Therefore, wide kinds of quantum systems can berealized by setting τ₁ at predetermined values.

[0181] The above examples are stellar fractal structures in which α=0and α=2 coexist. In contrast, a simple fractal structure of α=1 is of alevel of the fractal dimension (D_(f)˜2.45), and a quantum chaoticsystem of α=0 and an integrable system can be connected via the simplefractal of α=1. FIGS. 24 and 25 show quantum level statistics in astellar fractal structure of (α₁, α₂)=(0, 1). FIGS. 26 and 27 showquantum level statistics in a stellar fractal structure of (α₁, α₂)=(1,2). FIGS. 24 and 26 show P(s), and FIGS. 25 and 27 show Δ₃ statistics.τ₁=10000 of FIGS. 24 and 25 is the same quantum chaotic system asτ₁=10000 of FIGS. 22 and 23. As shown in FIGS. 24 and 25, as τ₁decreases, the quantum level statistics changes from that of GUEdistribution to that of Poisson distribution. However, even in τ₁ =0 ofFIGS. 24 and 25, i.e., even in the simple fractal of α=1, it does notexhibit complete Poisson distribution. Let us change the eyes to FIGS.26 and 27. τ₁=10000 of FIGS. 26 and 27 is the same as τ₁=0 of FIGS. 24and 25, and it is the quantum level statistics in the simple fractalstructure of α=1. As τ₁ decreases, the quantum level statisticsapproaches that of Poisson distribution, and in a region with small τ₁,it asymptotically approaches that of Poisson distribution. Of course,τ₁=0 of FIGS. 26 and 27 is the quantum level statistics in the simplefractal of α=2, and it is the same as τ₁=0 of FIGS. 22 and 23.

[0182] In correspondence with the change from the quantum chaotic systemdescribed by GOE distribution, observed under no magnetic field, to theintegrable quantum system described by Poisson distribution, it isobserved under the random magnetic field that the quantum chaotic systemdescribed by GUE distribution changes to an integrable quantum systemdescribed by Poisson distribution. That is, it has been confirmed thatwidely various kinds of quantum chaotic systems from quantum chaoticsystem described by GOE distribution to quantum chaotic systemsdescribed by GUE distribution can be obtained.

[0183] Fifth Embodiment

[0184] In the fifth embodiment, fractal dimensions suitable forcontrolling correlative electron systems by stellar fractal structureswill be explained.

[0185] (1) Formation of a Stellar Fractal Structure

[0186] A method of forming the star-shaped fractal structure accordingto the fifth embodiment is the same as the method of forming the stellarfractal structure according to the first embodiment. As growthconditions, however, while fixing τ=5000 and τ₂=10000, (α₁, α₂) waschanged variously within the range satisfying α₁<α₂ for conductingnumerical experiment of growth.

[0187] Regarding methods of forming a single fractal structure by adielectric breakdown model, it is known that changes of a invite changesof the fractal dimension D_(f) of the fractal structure to be formed.Fractal dimensions having obtained by simulation are shown in FIG. 28(see Literature (34)). As apparent from FIG. 28, as α increases, D_(f)decreases. When α<0.5, D_(f)<2.7 is obtained. When α>1, D_(f)<2.3 isobtained.

[0188] (2) Correlative Electron System in a Stellar Fractal Structure

[0189] An electron system is defined on the stellar fractal structuredefined in (1). A review is made about the lattice point

r _(p)=(i _(1,p) ,i _(2,p) ,i _(3,p))∈T _(n)  (77)

[0190] that is the origin of T_(n). In Equation (9), p=0, 1, 2, . . . ,n. An operator Ĉ_(p,σ) ^(†) for generating an electron of a spin σ isdefined at the lattice point r_(p)∈T_(n). Of course, there is theanticommutative relation

{ĉ _(p,σ) ,ĉ _(q,ρ) ^(†)}=δ_(p,q)δ_(σ,ρ)  (61)

[0191] Here is defined a single-band Hubbard Hamiltonian Ĥ of theelectron system as follows. $\begin{matrix}{\hat{H} = {{t{\sum\limits_{i,j,\sigma}{\lambda_{i,j}{\hat{c}}_{i,\sigma}^{\dagger}{\hat{c}}_{j,\sigma}}}} + {U{\sum\limits_{j}{{\hat{n}}_{j, \uparrow}{\hat{n}}_{j, \downarrow}}}}}} & (79)\end{matrix}$

[0192] Assuming that electrons are movable only among nearest-neighborsites, the following is employed as λ_(p,q). $\begin{matrix}{\lambda_{p,q} = \{ \begin{matrix}1 & {{{when}\quad {{r_{p} - r_{q}}}} = 1} \\0 & {otherwise}\end{matrix} } & (80)\end{matrix}$

[0193] Additionally, the spin σ electron density operator of the j-thsite, {circumflex over (n)}_(j,σ)=ĉ_(j,σ) ^(†)ĉ_(j,σ), and their sum,{circumflex over (n)}_(j)=Σ_(σ){circumflex over (n)}_(j,σ), are defined.

[0194] For the purpose of defining a temperature Green's function, hereis introduced a grand canonical Hamiltonian {circumflex over(K)}=Ĥ−μ{circumflex over (N)} where {circumflex over(N)}=Σ_(j){circumflex over (n)}_(j). In the half-filled taken here,chemical potential is μ=U/2. The half-filled grand canonical Hamiltoniancan be expressed as $\begin{matrix}{\hat{K} = {{t{\sum\limits_{i,j,\sigma}{\lambda_{j,i}{\hat{t}}_{j,i,\sigma}}}} + {{U/2}{\sum\limits_{i}( {{\hat{u}}_{i} - 1} )}}}} & (81)\end{matrix}$

[0195] Operators {circumflex over (t)}_(j,i,σ), ĵ_(j,i,σ), û_(i) and{circumflex over (d)}_(i,σ) are defined beforehand as

{circumflex over (t)} _(j,i,σ) =ĉ _(j,σ) ^(†) ĉ _(i,σ) +ĉ _(j,σ) ^(†) ĉ_(i,σ)  (82)

ĵ _(j,i,σ) =ĉ _(j,σ) ^(†) ĉ _(i,σ) −ĉ _(j,σ) ^(†) ĉ _(i,σ)  (83)

û _(i) =ĉ _(i,↑) ^(†) ĉ _(i,↑) ĉ _(i,↑) ĉ _(i,↓) ^(†) ĉ _(i,↓) +ĉ _(i,↑)^(†) ĉ _(i,↑) ^(†) ĉ _(i,↓) ĉ _(i,↓) ^(†)  (85)

{circumflex over (d)} _(i,σ) =ĉ _(i,σ) ^(†) ĉ _(i,σ) −ĉ _(i,σ) ĉ _(i,σ)^(†)  (86)

[0196] If the temperature Green function is defined for operators Â and{circumflex over (B)} given, taking τ as imaginary time, it is asfollows.

<Â;{circumflex over (B)}>=−∫ ₀ ^(β) dτ<T _(τ) Â(τ){circumflex over(B)}>e ^(iw) ^(_(n)) ^(t)  (86)

[0197] The on-site Green function

G _(j,σ)(ω _(n))=<ĉ _(j,σ) ;ĉ _(j,σ) ^(†)>  (87)

[0198] is especially important.

[0199] Imaginary time development of the system is obtained by theHeisenberg equation $\begin{matrix}{{\frac{\quad}{t}{\hat{A}(\tau)}} = \lbrack {\hat{K},\hat{A}} \rbrack} & (88)\end{matrix}$

[0200] As the equation of motion of the on-site Green function,$\begin{matrix}{{\quad \omega_{n}{\langle{{\hat{c}}_{j,\sigma};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}} = {1 + {t{\sum\limits_{p,j}{\lambda_{p,j}{\langle{{\hat{c}}_{p,\sigma};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}} + {\frac{U}{2}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}} & (89)\end{matrix}$

[0201] is obtained. Then, the approximation shown below is introduced,following Gros ((35) C. Gros, Phys. Rev. B50, 7295(1994)). If the site pis the nearest-neighbor site of the site j, the resolution

<ĉ _(p,σ) ;ĉ _(j,σ) ^(†) >→t<ĉ _(p,σ) ĉ _(p,σ) ^(†) ><ĉ _(j,σ) ĉ _(j,σ)^(†)>  (90)

[0202] is introduced as the approximation. This is said to be exact incase of infinite-dimensional Bethe lattices, but in this case, it isonly within approximation. Under the approximation, the followingequation is obtained. $\begin{matrix}{{{( {{\quad \omega_{n}} - {t^{2}\Gamma_{j,\sigma}}} )G_{j,\sigma}} = {1 + {\frac{U}{2}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}}{where}} & (91) \\{\Gamma_{j,\sigma} = {\sum\limits_{p}{\lambda_{p,j}G_{p,\sigma}}}} & (92)\end{matrix}$

[0203] was introduced. To solve the equation, obtained, <{circumflexover (d)}_(j,−σ)ĉ_(j,σ);ĉ_(j,σ) ^(†)> has to be analyzed. In case ofhalf-filled models, this equation of motion is $\begin{matrix}{{\quad \omega_{n}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}} = {{\frac{U}{2}G_{j,\sigma}} - {2t{\sum\limits_{p}{\lambda_{p,\quad j}{\langle{{{\hat{j}}_{p,j,{- \sigma}}{\hat{c}}_{j,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}} + {t{\sum\limits_{p}{\lambda_{p,j}{\langle{{{\hat{d}}_{j,{- \sigma}}{\hat{c}}_{p,\sigma}};{\hat{c}}_{j,\sigma}^{\dagger}}\rangle}}}}}} & (93)\end{matrix}$

[0204] Here again, with reference to the Gros theory, approximation isintroduced. It is the following translation.

<ĵ _(p,j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†) >→tG _(p,−σ) <{circumflex over (d)}_(j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†)>  (94)

<{circumflex over (d)} _(j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†) >→tG _(p,σ)<{circumflex over (d)} _(j,−σ) ĉ _(j,σ) ;ĉ _(j,σ) ^(†)>  (95)

[0205] By executing this translation, the following closed equation isobtained. $\begin{matrix}{{( {{\quad \omega_{n}} - {t^{2}\Gamma_{j,\sigma}}} )G_{j,\quad \sigma}} = {1 + {\frac{( {U/2} )^{2}}{{\quad \omega_{n}} - {t^{2}\Gamma_{j,\sigma}} - {2t^{2}\Gamma_{j,{- \sigma}}}}G_{j,\sigma}}}} & (96)\end{matrix}$

[0206] Here is assumed that there is no dependency on spin. That is,assuming

G _(j) =G _(j,↑) =G _(j,↓)  (97)

[0207] the following calculation is executed. This is because, whenanalytic continuation is conducted as iω_(n)→ω+iδ for small δ,

D _(j)(ω)=−ImG _(j)(ω+iδ)  (98)

[0208] becomes the local density of states of the site j, and$\begin{matrix}{{D(\omega)} = {{- \frac{1}{n + 1}}{\sum\limits_{j}{D_{j}(\omega)}}}} & (99)\end{matrix}$

[0209] becomes the density of states of the system. For later numericalcalculation of densities of states, δ=0.0001 will be used.

[0210] In the following calculation, t=1 and U=7 are fixed. For adetailed review of spatial changes inside a stellar fractal structure,spatial changes of the local density of states D_(j) (ω=0) on the Fermienergy are analyzed. When the distance from the growth origin (0, 0, 0)is r, the set of growth points distant from the growth origin by r tor+dr is written as U_(s)(r), and its original number is written asM_(s)(r). Then, D_(ave)(r) is defined as follows. $\begin{matrix}{{D_{ave}(r)} = {\frac{1}{M_{s}(r)}{\sum\limits_{j\quad \in \quad {U_{s}{(r)}}}{D_{j}(0)}}}} & (100)\end{matrix}$

[0211] This amount is the average value of local densities of states onthe Fermi energy on the spherical surface with the distance r from thegrowth origin. In this calculation, dr=1 is used.

[0212]FIG. 29 shows M_(s)(r) under α₂=2 and α=0, 0.2, 0.4, 0.6 and 1.FIG. 30 shows M_(s)(r) under α₂=1 and α₁=0.2, 0.4 and 0.6. FIG. 31 showsM_(s)(r) under α₂=0.6 and α₁=0, 0.2 and 0.4. FIG. 32 shows D_(ave)(r)under α₂=2 and α₁=0, 0.2, 0.4 and 0.6. FIG. 33 shows D_(ave)(r) underα₂=1 and α₁=0, 0.2, 0.4 and 0.6. FIG. 34 shows D_(ave)(r) under α₂=0.6and α₁=0, 0.2 and 0.4.

[0213] It is confirmed from FIGS. 32, 33 and 34 that, in the region withsmall r, that is, in the region of the somatic fractal region,D_(ave)(r) takes finite values, the system behaves as a metal. As rincreases, D_(ave)(r) decreases and approaches zero asymptotically. Thatis, Mott insulator transformation occurs. To obtain a sufficient changeof densities of states, a value around α₁<0.4 inside and a value aroundα₂>1 outside are required, and a fractal dimension of D_(f)>2.7 insideand a fractal dimension of D_(f)<2.3 outside are required.

[0214] Sixth Embodiment

[0215] In the sixth embodiment, fractal dimensions suitable forcontrolling magnetization curves by stellar fractal structures will beexplained.

[0216] (1) Formation of a Stellar Fractal Structure

[0217] A method of forming the star-shaped fractal structure accordingto the sixth embodiment is the same as the method of forming the stellarfractal structure according to the first embodiment. As growthconditions, however, while fixing τ₁=5000 and τ₂=10000, (α₁, α₂) waschanged variously within the range satisfying α₁<α₂ for conductingnumerical experiment of growth.

[0218] (2) Ferromagnetic Transition in a Stellar Fractal Structure

[0219] A spin system describing a ferromagnetic structure is defined onthe stellar fractal structure defined in (1). A review is made about thelattice point

r _(p)=(i _(1,p) ,i _(2,p) ,i _(3,p))∈T _(n)  (101)

[0220] that is the origin of T_(n). In Equation (101), p=0, 1, 2, . . ., n. By placing a spin on a lattice site r_(p)∈T_(n), here is taken aspin system that can be described by the following Hamiltonian.$\begin{matrix}{H = {- {\sum\limits_{p,q}{J_{p,q}{S_{p} \cdot S_{q}}}}}} & (102)\end{matrix}$

[0221] S_(p) is the spin at the p site. A natural model of spin-spininteraction J_(p,q) is: $\begin{matrix}{J_{p,q} = \{ \begin{matrix}1 & {{{when}\quad {{r_{p} - r_{q}}}} = 1} \\0 & {otherwise}\end{matrix} } & (103)\end{matrix}$

[0222] That is, spin-spin interaction exists only betweennearest-neighbor sites. For the purpose of calculating spontaneousmagnetization M at a finite temperature T, statistical mechanics of anequilibrium system is introduced. A partition function Z is defined asfollows. $\begin{matrix}{Z = {\sum\limits_{\{ s_{p}\}}^{{- H}/T}}} & (104)\end{matrix}$

[0223] where {S_(p)} in the symbol of the sum pertains to the sumregarding all spin states. Spontaneous magnetization is defined as thestatistical average of spins as follows. $\begin{matrix}{M = {\frac{1}{n + 1}{\sum\limits_{p = 1}^{n}{\langle S_{p}\rangle}}}} & (105)\end{matrix}$

[0224] where the expected value <S_(p)> is $\begin{matrix}{{\langle S_{p}\rangle} = {\frac{1}{Z}{\sum\limits_{\{ S_{p}\}}{S_{p}^{{- H}/T}}}}} & (106)\end{matrix}$

[0225] and n+1 is the total number of spins. M is generally a vectorquantity in a spin space, but its absolute value M=|M| is calculated.

[0226] Here is made a review about an Ising model. In an Ising model,only two states of

S _(P)=1 or −1  (107)

[0227] can exist. Let a mean field approximation be introduced into theIsing model. Spontaneous magnetization of the p-th site is written asμ_(p). In this system, since the molecular field varies with site, letit written as {overscore (μ)}_(p). As an assumption of the mean fieldapproximation, here is employed one whose molecular field can be writtenby spontaneous magnetization of the nearest-neighbor site as$\begin{matrix}{{\overset{\_}{\mu}}_{p} = {\sum\limits_{q}{J_{p,q}\mu_{q}}}} & (108)\end{matrix}$

[0228] This assumption simplifies the foregoing Hamiltonian to$\begin{matrix}{H_{MF} = {- {\sum\limits_{P = 0}^{n}{{\overset{\_}{\mu}}_{p}\sigma_{p}}}}} & (109)\end{matrix}$

[0229] A self-consistent equation ensuring spontaneous magnetizationobtained by using a partition function by the simplified Hamiltonianbecomes μ_(p) results in

μ_(p)=tan h(β{overscore (μ)}_(p))  (110)

[0230] and by numerically solving this equation, spontaneousmagnetization of the system $\begin{matrix}{M_{Ising} = {\frac{1}{n + 1}{\sum\limits_{j = 0}^{n}\mu_{j}}}} & (111)\end{matrix}$

[0231]FIG. 35 shows spontaneous magnetization in case of α₁=0 fixed andα₂ changed. Here, α₂=0 corresponds to spontaneous magnetization in thesame structure as the simple fractal of α=0. As compared with thespontaneous magnetization on that structure, as α₂ increases,spontaneous magnetization changes to one peculiar to stellar fractalstructures, of which the magnetization curve changes in direction ofinclination. In this case, favorable spontaneous magnetization isobtained under around α≧0.6.

[0232]FIG. 36 shows spontaneous magnetization in case of α₁=0.2 fixedand α₂ changed. Here, α₂=0.2 corresponds to spontaneous magnetization inthe same structure as the simple fractal of α=0.2. As compared with thespontaneous magnetization on that structure, as α₂ increases,spontaneous magnetization changes to one peculiar to stellar fractalstructures, of which the magnetization curve changes in direction ofinclination. In this case, favorable spontaneous magnetization isobtained under around α₂≧1.

[0233]FIG. 37 shows spontaneous magnetization in case of α₁=0.4 fixedand α₂ changed. Here, α₂=0.4 corresponds to spontaneous magnetization inthe same structure as the simple fractal of α=0.4. As compared with thespontaneous magnetization on that structure, as α₂ increases,spontaneous magnetization changes to one peculiar to stellar fractalstructures, of which the magnetization curve changes in direction ofinclination. In this case, favorable spontaneous magnetization isobtained under around α₂=≧1.

[0234]FIG. 38 shows spontaneous magnetization in case of α₂=2 fixed andα₁ changed. Here, α₁=2 corresponds to spontaneous magnetization in thesame structure as the simple fractal of α=2. As compared with thespontaneous magnetization on that structure, as α₂ decreases,spontaneous magnetization changes to one peculiar to stellar fractalstructures, of which the magnetization curve changes in direction ofinclination. In this case, favorable spontaneous magnetization isobtained under around α₁≦0.4.

[0235]FIG. 39 shows spontaneous magnetization in case of α₂=1 fixed andα₁ changed. Here, α₁=1 corresponds to spontaneous magnetization in thesame structure as the simple fractal of α=1. As compared with thespontaneous magnetization on that structure, as α₂ decreases,spontaneous magnetization changes to one peculiar to stellar fractalstructures, of which the magnetization curve changes in direction ofinclination. In this case, favorable spontaneous magnetization isobtained under around α₁<0.2.

[0236]FIG. 40 shows spontaneous magnetization in case of α₂=0.6 fixedand α₁ changed. Here, α₁=0.6 corresponds to spontaneous magnetization inthe same structure as the simple fractal of α=0.6. As compared with thespontaneous magnetization on that structure, as α₂ decreases,spontaneous magnetization changes to one peculiar to stellar fractalstructures, of which the magnetization curve changes in direction ofinclination. In this case, favorable spontaneous magnetization isobtained under around α₁≦0.2.

[0237] Generalizing the above aspects of spontaneous magnetization, whena stellar structure satisfies the condition of approximately D_(f)>2.7as the fractal dimension of the somatic region inside or approximatelyD_(f)<2.3 as the fractal dimension of the dendritic region outside, themagnetization curve of the stellar fractal structure exhibitsdistinctive behaviors, and enables favorable control.

[0238] Seventh Embodiment

[0239] A method for forming a stellar fractal structure according to theseventh embodiment is the same as the method of forming the stellarfractal structure according to the fist embodiment. As growthconditions, however, while fixing τ₁ =5000 and τ₂=10000, variouscombinations of (α₁,α₂) satisfying α₁<α₂, particularly, 0, 0.2, 0.4,0.6, 1 and 2 as α₁ and α₂, were used for conducting numerical experimentof growth.

[0240] (2) Electron System on a Stellar Fractal Structure

[0241] In the same manner as that shown in Equations (44) through (59)of the third embodiment, a quantum system of one particle is defined onthe stellar fractal defined in (1).

[0242] Since the stellar fractal structure analyzed here is one obtainedby growth experiment of n=10000, this quantum system includes n+1=10001eigenstates. Based on energy eigenvalues concerning 1501 states from the510-th to 2001-th eigenstates from the ground state, among thoseeigenstates, the following quantum level statistics was calculated.FIGS. 41 and 42 show quantum level statistics in stellar fractalstructures of (α₁,α₂)=(0, x) where x=0, 0.2, 0.4, 0.6, 1 and 2.

[0243]FIG. 41 shows P(s) and FIG. 42 shows Δ₃ statistics. In case ofα₂=0, the structure is the same as the simple fractal of α=0, and thefractal dimension is near 3 (D_(f)˜2.91). Therefore, the system behavesas a quantum chaotic system. In this case, the quantum level statisticsis that of GOE distribution. As α₂ increases, the quantum levelstatistics goes apart from that of GOE distribution toward Poissondistribution. However, even when it reaches α₂=2, a large differencefrom Poisson distribution still remains.

[0244]FIG. 43 shows Δ₃ statistics in stellar fractal structures of(α₁,α₂)=(0.2, x) where x=0.2, 0.4, 0.6, 1 and 2. In case of α₂=0.2, thestructure is the same as the simple fractal of α=0.2, and behavessubstantially as a quantum chaotic system. As α₂ increases, the quantumlevel statistics goes apart from that of GOE distribution toward Poissondistribution. FIG. 44 shows Δ₃ statistics in stellar fractal structuresof (α₁,α₂)=(0.4, x) where x=0.4, 0.6, 1 and 2. In case of α₂=0.4, thestructure is the same as the simple fractal of α=0.4, and it is as farfrom that of GOE distribution as a level to be no more regarded as aquantum chaotic system. As α₂ increases, the quantum level statisticssimilarly goes apart from that of GOE distribution toward Poissondistribution.

[0245]FIGS. 45 and 46 show quantum level statistics in stellar fractalstructures of (α₁,α₂)=(x, 2) where x=0, 0.2, 0.4, 0.6, 1 and 2. FIG. 45shows P(s) and FIG. 46 shows Δ₃ statistics. In case of α₁=0, it is thesame as α₂=2 of FIGS. 41 and 42. In case of α₁=2, the structure is thesame as the simple fractal structure of α=2, and the fractal dimensionis near 2 (D_(f)˜2.16). Therefore, the system behaves as an integrablesystem. As α₁ decreases, the quantum level statistics goes apart fromthat of Poisson distribution toward that of GOE distribution.

[0246]FIG. 47 shows Δ₃ statistics in stellar fractal structures of(α₁,α₂)=(x, 1) where x=0, 0.2, 0.4, 0.6 and 1. FIG. 48 shows Δ₃statistics in stellar fractal structures of (α₁,α₂)=(x, 0.6) where x=0,0.2, 0.4 and 0.6. As a, decreases, the quantum level statistics goesapart from that of Poisson distribution toward that of GOE distribution.

[0247] For quantitative evaluation of the above-reviewedcontrollability, the Berry-Robnik parameter ρ is used ((38) M. V. Berryand M. Robnik, J. Phys. A (Math. Gen.) 17, 2413,(1984)). First, when{overscore (ρ)}=1−ρ, $\begin{matrix}{{{P_{2}( {s,\rho} )} = {{\rho^{2}^{{- \rho}\quad s}{{erf}( \frac{\sqrt{\pi}\overset{\_}{\rho}\quad s}{2} )}} - {( {{2\rho \quad \overset{\_}{\rho}} + \frac{\pi \quad {\overset{\_}{\rho}}^{3}s}{2}} )^{{{- \rho}\quad s} - {\pi \quad {\overset{\_}{\rho}}^{2}{s^{2}/4}}}}}}{{{is}\quad {introduced}},{where}}} & (112) \\{{{erf}(x)} = {\frac{2}{\sqrt{\pi}}{\int_{x}^{\infty}\quad {{\tau}\quad ^{- \tau^{2}}}}}} & (113)\end{matrix}$

[0248] was used. This function P₂(s, ρ) coincides with P(s) of Poissondistribution under ρ=1, and coincides with P(s) of GOE distributionunder ρ=0. That is, by changing ρ from 0 to 1, quantum level statisticsfrom those of quantum chaotic systems to those of integrable systems canbe interpolated. The Berry-Robnik parameter is the value of ρ obtainedby optimum approximation of P(s) obtained by numerical calculation withP₂(s, ρ) shown above. Within the range of semiclassical approximation, ρis the ratio in volume of regular regions (integrable systems andregions capable of perturbing development therefrom) in a phase space.

[0249]FIG. 49 shows the Berry-Robnik parameter ρ in stellar fractalstructures. (α₁,α₂)=(x, 2) is the Berry-Robnik parameter in the casewhere α₁ is placed on the abscissa while fixing α₂=2. (α₁,α₂)=(0, x) isthe Berry-Robnik parameter in the case where α₁ is placed on theabscissa while fixing α₂=0. It is apparent from FIG. 49 that widelyvarious quantum systems from quantum chaotic systems to integrablesystems can be realized by setting (α₁,α₂) to predetermined values.

[0250] Although the invention has been explained above by way ofspecific embodiments, the invention is not limited to those embodimentsbut envisages various changes or modifications based on the technicalconcept of the invention.

[0251] As described above, according to the invention, by growthconditions of a fractal structure with time, it is possible to obtain afractal structure including a plurality of regions different in fractaldimension characterizing the self-similarity, thereby to modulate andcontrol the dimensionality of a material by using a design methodexceeding the conventional simple fractal property. Then, in thesefractal structures, by adjusting the ratio in volume of a plurality ofregions, natures of various phase transition occurring in fractalstructures can be controlled. Additionally, by optimization of thefractal dimension, the controllability can be improved.

1. A fractal structure comprising a plurality of regions different infractal dimension characterizing the self-similarity.
 2. The fractalstructure according to claim 1 wherein the nature of phase transitionoccurring in the fractal structure is controlled by adjusting the ratioin volume of said plurality of regions relative to the entire volume ofthe fractal structure.
 3. The fractal structure according to claim 1wherein electron-to-electron correlation of an interactive electronsystem is controlled by adjusting the ratio in volume of said pluralityof regions relative to the entire volume of the fractal structure. 4.The fractal structure according to claim 1 wherein the magnetizationcurve of ferromagnetic phase transition is controlled by adjusting theratio in volume of said plurality of regions relative to the entirevolume of the fractal structure.
 5. The fractal structure according toclaim 1 wherein the nature of chaos appearing in the fractal structureis controlled by adjusting the ratio in volume of said plurality ofregions relative to the entire volume of the fractal structure.
 6. Thefractal structure according to claim 1 wherein quantum chaos in theelectron state is controlled by adjusting the ratio in volume of saidplurality of regions relative to the entire volume of the fractalstructure.
 7. The fractal structure according to claim 1 wherein quantumchaos in the electron state is controlled by addition of a magneticimpurity.
 8. The fractal structure according to claim 6 wherein quantumchaos in the electron state is controlled by addition of a magneticimpurity.
 9. The fractal structure according to claim 1 wherein saidfractal structure includes: a first region having a first fractaldimension and forming a core; and one or more second regions surroundingsaid first region and having a second fractal dimension lower than saidfirst fractal dimension.
 10. The fractal structure according to claim 1wherein said first region and said second region exhibit a stellar shapeas a whole.
 11. The fractal structure according to claim 9 satisfyingD_(f1)>2.7 and D_(f2)<2.3 where D_(f1) is said fractal dimension andD_(f2) is said second fractal dimension.
 12. The fractal structureaccording to claim 9 satisfying 2.7<D_(f1)≦3 and 1≦D_(f2)<2.3 whereD_(f1) is said fractal dimension and D_(f2) is said second fractaldimension.
 13. The fractal structure according to claim 9 satisfying2.9≦D_(f1)≦3 and 1≦D_(f2)<2.3 where D_(f1) is said fractal dimension andD_(f2) is said second fractal dimension.
 14. A method of forming afractal structure having a plurality of regions different in fractaldimension characterizing the self-similarity, comprising: growing afractal structure from one or more origins, and changing growthconditions with time in the growth process thereof such that differentfractal dimensions are obtained.
 15. The method of forming a fractalstructure according to claim 14 wherein there ire used growth conditionsensuring the first fractal dimension to be made from the growth startpoint of tine until a first point of time, and growth conditionsensuring a second fractal dimension lower than the first fractaldimension to be made from the first point of time to a second point oftime.
 16. The method of forming a fractal structure according to claim15 satisfying D_(f1)>2.7 and D_(f2)<2.3 where D_(f1) is said fractaldimension and D_(f2) is said second fractal dimension.
 17. The method offorming a fractal structure according to claim 15 satisfying2.7<D_(f1)≦3 and 1≦D_(f2)<2.3 where D_(f1) is said fractal dimension andD_(f2) is said second fractal dimension.
 18. The method of forming afractal structure according to claim 15 satisfying 2.9≦D_(f1)≦3 and1≦D_(f2)<2.3 where D_(f1) is said fractal dimension and D_(f2) is saidsecond fractal dimension.